Deep-Sea Bubbles

Deep sea vents can emit harmful gases, such as hydrogen sulphide. Since these bubbles are small, they shrink once they leave the vent, as the gases dissolve into the ocean.

An autonomous submarine takes a photograph of these bubbles that looks like this graph sketch. 

deep-sea bubbles

A scientist discovers that the surface of the bubbles visible in this photograph can be represented by the curve:

\[\sin\frac{y^{3}}{100} + e^{- x}y + x^{2} = 5\ ,\ y > 5\]

By first using implicit differentiation to show that

\[\frac{\text{dy}}{\text{dx}} = \frac{100(y – 2e^{x}x)}{3e^{x}y^{2}\cos\frac{y^{3}}{100} + 100}\]

Show that at the turning points on the curve, where \(\frac{\text{dy}}{\text{dx}} = 0\), \(y = 2xe^{x}\)

[8 marks]

Filling a Weather Balloon

The height (h km) that a weather balloon can reach is related to the volume (v m₃) of helium in it at sea level by the equation:

\[h = \frac{8v^{2}}{5} – \frac{32v^{3}}{255}\ ,\ v \leq 12\]

a) Find the volume of helium required to achieve the maximum height and state this height.

[5 marks]

Storing Sequestrates

A manufacturer produces a tank for storing liquid CO2 underground.

The tank is modelled in the shape of a hollow vertical circular cylinder closed with a flat lid at the top and a hemispherical shell at the bottom. 

The walls of the tank are assumed to have negligible thickness.

The cylinder has a radius r metres and height h metres and the hemisphere has radius r metres.

The volume of the tank is 6 m3.

a) Show that, according to the model, the surface area of the tank, in m2 is given by

\[\frac{12}{r} + \frac{5}{3}\pi r^{2}\]

[4 marks]

The manufacturer needs to minimise the surface area of the tank, to minimise costs. 

b) Find, using calculus, the radius of the tank for which the surface area is a minimum.

[4 marks]

c) Find the surface area of the tank for this radius, giving your answer to the nearest integer.

[2 marks]